A138099 Irregular triangle read by rows: T(n,k) = k + floor(n/2), 1 <= k <= ceiling(n/2).

1, 2, 2, 3, 3, 4, 3, 4, 5, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 9, 10, 11, 12, 13

Offset: 1

Author

Paul Curtz, May 03 2008

Keywords

• nonn
• easy
• tabf

Comments

Previous name was: Base for 8 X 32 Janet periodic table of elements. Ordered energy levels of atoms.

Examples

			Triangle begins:
1;
2;
2, 3;
3, 4;
3, 4, 5;
4, 5, 6;
4, 5, 6, 7;
5, 6, 7, 8;
...
		

Programs

Magma

lst:=[]; for n in [1..16] do for k in [1..Ceiling(n/2)] do Append(~lst, k+Floor(n/2)); end for; end for; lst; // Arkadiusz Wesolowski, Jun 15 2014

Mathematica

Table[k+Floor[n/2],{n,20},{k,Ceiling[n/2]}]//Flatten (* Harvey P. Dale, Aug 15 2017 *)

Formula

a(n) = n - floor((1/4)*(floor(sqrt(4*n-3))-1)^2). - Mircea Merca, Feb 05 2012

From Benedict W. J. Irwin, May 14 2016: (Start)

Conjecture: a(n) = HookSum(T(n))-HookSum(T(n-1)), where HookSum(A) is the sum of all hooks across a tableau A, and T(n) are the spiraling sequence of tableaux:

o oo ooo

Empty, o, oo, oo, oo, oo , ooo , ooo, ooo, ooo, ...

o oo ooo ooo ooo ooo ooo

(End)

Extensions

Name changed and more terms added by Arkadiusz Wesolowski, Jun 15 2014